Atlas is commonly depicted holding the world up: a great globe on his
shoulders, and he in some agonized
contrapposto position to support it, like an over-ambitious shot-putter.
As Webster 1913 says, the use of this image in Gerhardus Mercator's book of maps was the origin of the term atlas in that sense. I don't know how long this image was around before then, but it might well have been contemporary (post-Columbus), rather than going back all the way to classical mythology.
When did people start believing the earth was spherical? It is ascribed to the Pythagoreans, but I don't know how widely accepted it was in ancient Greece, or whether it was just esoteric theory then.
In the original myth, Atlas stood upon the earth, and on his shoulders carried the heavens, or more accurately carried the pillars that held up the heavens. He was Mount Atlas, or the Atlas Mountains in what is now Morocco.
One story says that Perseus met him there and turned him to stone with the head of Medusa.
But in a different version he was alive when he held up the heavens. Hercules came to him seeking the Golden Apples of the Hesperides. He agreed to hold up the heavens while Atlas went off to fetch the apples. Atlas of course could hardly believe his luck. Hercules, being a strong man, was not exactly known for brains. So Atlas went off chuckling, got the apples, and planned to tell Hercules, sorry mate, you're it now, I'm off to disport with the nymphs. (He did at some point father Calliope and the Pleiades.)
So Hercules said yeah, okay... er, excuse me, I've got a bit of an itchy shoulder. This pad is slipping. I just need to adjust it. Do you mind just holding the heavens up again while I do it?
Hint: What did I tell you about strong men? You've never heard of the Twelve Labours of Atlas, have you?
A male figure holding up the roof of a building is called an atlantid (or telamon), as the female is called a caryatid.
The Greek root is tla- 'to bear'. Cognate Germanic words are the archaic English thole 'to bear, to suffer patiently' and German geduldig 'patient'. Also the Latin root tol- as in tolerate, and the participle tlat- which was suppletive for fero 'I bear': but the stem got its consonants simplified to lat-, so you got pairs like transfero 'I carry across', translatum 'carried across'.